# 1 Example 2 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,  ] into 6 subintervals. Solution Observe that the function.

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1 Example 2 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,  ] into 6 subintervals. Solution Observe that the function with g(0)=1 is continuous at x=0 because Note that P = {0,  /6,  /3,  /2, 2  /3, 5  /6,  } with each subinterval of width  /6. The six subintervals are: [0,  /6 ], [  /6,  /3], [  /3,  /2], [  /2, 2  /3], [2  /3, 5  /6] and [5  /6,  ]. Then The L k are the heights of the rectangles used to approximate this definite integral. In the Left Endpoint Rule the L k are the values of g on the left endpoints of the six subintervals: g(0), g(  /6), g(  /3), g(  /2), g(2  /3), g(5  /6). 

2 In the Right Endpoint Rule the L k are the values of g on the right endpoints of the six subintervals: g(  /6), g(  /3), g(  /2), g(2  /3), g(5  /6), g(  ). In the Midpoint Rule the L k are the values of g on the midpoints of the six subintervals: g(  /12), g(  /4), g(5  /12), g(7  /12), g(3  /4), g(11  /12). In the Trapezoid Rule the L k are the averages of the values of g on the endpoints of each of the six subintervals: ½[g(0)+ g(  /6)], ½[g(  /6)+ g(  /3)], ½[g(  /3)+ g(  /2)], ½[g(  /2)+ g(2  /3)], ½[g(2  /3)+ g(5  /6)], ½[g(5  /6)+ g(  )]. Therefore, the Lower Riemann sum coincides with the estimate of the Right Endpoint Rule and the Upper Riemann sum coincides with estimate of the Left Endpoint Rule. Since the function g is decreasing on [0,1], it has its maximum value at the left endpoint of each subinterval and its minimum value at the right endpoint of each subinterval. The values of the L k are summarized in the table on the next slide The six subintervals are: [0,  /6 ], [  /6,  /3], [  /3,  /2], [  /2, 2  /3], [2  /3,5  /6], [5  /6,  ].

3 The Left Endpoint Rule and Upper Riemann Sum give the same estimate: The Right Endpoint Rule and Lower Riemann Sum give the same estimate: The Midpoint Rule gives the estimate: The Trapezoid Rule gives the estimate: Left Endpoint Rule Right Endpoint Rule Midpoint Rule Trapezoid Rule Lower Riemann Sum Upper Riemann Sum L1L2L3L4L5L6L1L2L3L4L5L6 g(0)=1 g(  /6) .955 g(  /3) .827 g(  /2) .637 g(2  /3) .414 g(5  /6) .191 g(  /6) .955 g(  /3) .827 g(  /2) .637 g(2  /3) .414 g(5  /6) .191 g(  )=0 g(  /12) .989 g(  /4) .900 g(5  /12) .738 g(7  /12) .527 g(3  /4) .300 g(11  /12) .090 ½(1+.955) ½(.955+.827) ½(.827+.637) ½(.637+.414) ½(.414+.191) ½(.191+0) g(  /6) .955 g(  /3) .827 g(  /2) .637 g(2  /3) .414 g(5  /6) .191 g(  )=0 g(0)=1 g(  /6) .955 g(  /3) .827 g(  /2) .637 g(2  /3) .414 g(5  /6) .191

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